Given the coordinates of a triangle $A(x_1,y_1)$, $B(x_2,y_2)$, $C(x_3,y_3)$, prove that the incentre and excenter of the triangle $\triangle ABC$ are harmonic conjugates with respect to the $A$ and $D$ lying on internal angle bisector passing through the the incentre and excenter.
In the image embedded for example, prove $AI/ID=AI_A/AD$.
I could only think of $AI/ID=(b+c)/a$. I couldn't prove the RHS equal to it. Any help is appreciated!